LARGE-EDDY SIMULATION OF TURBULENCE–RADIATION

INTERACTIONS IN A TURBULENT PLANAR CHANNEL FLOW

Ankur Gupta, Michael F. Modest ∗ and Daniel C. Haworth

Department of Mechanical and Nuclear Engineering,

The Pennsylvania State University

University Park, PA 16802, USA

ABSTRACT

Large-eddy simulation (LES) has been performed for planar turbulent channel flow

between two infinite, parallel, stationary plates. The capabilities and limitations of the

LES code in predicting correct turbulent velocity and passive temperature field statistics

have been established through comparison to DNS data from the literature for nonreacting

cases. Mixing and chemical reaction (infinitely fast) between a fuel stream and an oxidizer

stream have been simulated to generate large composition and temperature fluctuations

in the flow; here the composition and temperature do not affect the hydrodynamics (one-

way coupling). The radiative transfer equation is solved using a spherical harmonics (P1)

method, and radiation properties correspond to a fictitiousgray gas with a composition-

and temperature-dependent Planck-mean absorption coefficient that mimics that of typi-

cal hydrocarbon-air combustion products. Simulations have been performed for different

optical thicknesses. In the absence of chemical reactions,radiation significantly modifies

∗Fellow ASME; Corresponding author; Email: mfmodest@psu.edu1

the mean temperature profiles, but temperature fluctuationsand turbulence–radiation inter-

actions (TRI) are small, consistent with earlier findings. Chemical reaction enhances the

composition and temperature fluctuations, and hence the importance of TRI. Contributions

to emission and absorption TRI have been isolated and quantified as a function of optical

thickness.

Keywords: Large-eddy Simulation, Thermal Radiation, Turbulence–Radiation Interactions,

Chemical Reaction, Channel Flow

1 INTRODUCTION

Most practical combustion systems involve turbulent flow and operate at high temperatures

where thermal radiation acts as an important mode of heat transfer. In such systems there are highly

nonlinear interactions between chemistry, turbulence, and thermal radiation. Accurate treatment

of chemistry, radiation and turbulence in turbulent reacting flow, even without interactions, are

quite challenging and complex on their own; therefore, the interactions between turbulence and

radiation (turbulence–radiation interactions, TRI) havetraditionally been ignored in the modeling

of turbulent reacting flows.

TRI arise due to highly nonlinear coupling between the fluctuations of temperature, radiative

intensity and the species concentrations. TRI take place over a wide range of length scales. A

complete treatment requires consideration of the full range of scales, which is computationally

intractable for flows with high Reynolds number, as is the case with chemically reacting turbulent

flows of practical interest. Numerical simulation of such systems requires either computing the

mean flow while modeling the effects of fluctuations at all scales (Reynolds-averaged approach),

or explicitly resolving the large scales while modeling theeffects of subfilter-scale fluctuations

(Large-Eddy Simulation - LES approach). LES is expected to be more accurate and general, since

the large, energy-containing, flow-dependent scales are captured explicitly and only the (presum-

ably) more universal small-scale dynamics require modeling. LES is also expected to capture phe-

2

nomena that are difficult to accommodate in Reynolds-averaged approaches, suchas large-scale

unsteadiness. There is a wide and rapidly growing body of evidence that demonstrates quanti-

tative advantages of LES in modeling studies of laboratory flames [1, 2] and in applications to

gas-turbine combustors [3–5], IC engines [6–8], and other combustion systems. LES, therefore,

promises to be accurate and computationally feasible tool for investigations of TRI in chemically

reacting, turbulent flows.

The importance of TRI has long been recognized [9–16]. Studies have shown that the effects

of TRI can be as significant as those of turbulence−chemistry interactions. TRI is known to result

in higher heat loss due to increased radiative emission, reduced temperatures and, consequently,

significant changes in key pollutant species (particularlyNOx and soot) in chemically reacting

turbulent flows. Faeth and Gore and coworkers [12–16] have shown that, with the inclusion of

TRI, radiative emission from a flame may be 50% to 300% higher (depending on the type of fuel)

than that expected with radiation but no TRI. With consideration of TRI Coelho [17] reported a

nearly 50% increase in radiative heat loss for a nonpremixedmethane-air turbulent jet flame; Tesse

et al. [18] reported a 30% increase in the radiative heat lossfor a sooty nonpremixed ethylene-air

turbulent jet flame. Modest and coworkers [19, 20] observed about 30% increase in the radiative

heat flux with TRI in their studies of nonpremixed turbulent jet flames. Wu et al. [21] conducted

direct numerical simulation (DNS) of an idealized one-dimensional premixed turbulent flame to

study TRI along with a high-order photon Monte Carlo scheme (of order commensurate with the

underlying DNS code) to calculate the radiative heat source. They isolated and quantified the var-

ious contributions to TRI for different optical thicknesses. Deshmukh et al. [22] applied thesame

approach to DNS of an idealized, statistically homogeneous, nonpremixed system with full consid-

eration of TRI. Frankel et al. [23] performed large-eddy simulation of an idealized nonpremixed jet

flame using the optically thin eddy approximation [24] and treating emission TRI through a filtered

mass density function (FMDF) approach. Coelho [25] surveyed various computational approaches

to account for TRI and reviewed TRI studies based on those approaches.

3

In this work large-eddy simulation with a P1 radiation modelis used to isolate and quantify TRI

effects for a range of optical thicknesses for nonreacting and reacting channel flow configurations.

The objectives are to provide fundamental physical insightinto TRI in chemically reacting flows

and to provide guidance for model development, and to establish the suitability of LES for inves-

tigating TRI in chemically reacting flows. The remainder of this paper is organized as follows:

Section 2 presents the underlying theory behind turbulent–radiation interactions; the problem for-

mulation is outlined in Section 3; results and discussion for the two configurations are presented

in Section 4, followed by conclusions in Section 5.

2 THERMAL RADIATION AND TURBULENCE–RADIATION INTERACTION S (TRI)

The radiative source term in the instantaneous energy equation is expressed as the divergence

of the radiative heat flux. For a gray medium,

∇ ·~qrad = 4κPσT4− κPG (1)

whereκP is the Planck-mean absorption coefficient,σ is the Stefan-Boltzmann constant, andG ≡∫4π IdΩ is the direction-integrated intensity, or incident radiation, which is obtained by solving

the radiative transfer equation (RTE) [26]. The radiative source term consists of an emission part

and an absorption part: these are the first and second terms onthe right-hand side of Eq. (1),

respectively.

TRI can be brought into evidence by taking the mean of Eq. (1):

〈∇ ·~qrad〉 = 4σ〈κPT4〉− 〈κPG〉 (2)

In the emission term, TRI appears as a correlation between the Planck-mean absorption coefficient

4

and the fourth power of temperature:〈κPT4〉 = 〈κP〉〈T4〉+ 〈κ′P(T4)′〉, where a prime denotes a

fluctuation about the local mean. The presence of TRI in the emission term is further manifested in

the temperature self-correlation (〈T4〉 , 〈T〉4). In the absorption term, TRI appears as a correlation

between the Planck-mean absorption coefficient and the incident radiation:〈κPG〉 = 〈κP〉〈G〉+

〈κ′PG′〉.

In the present study, we explore the effects of resolved-scale fluctuations on mean and rms

temperature profiles and their contribution to TRI using LES. With ˜ denoting a filtered (resolved-

scale) value, the principal quantities examined are the mean (〈T〉) and rms (〈T′2〉) temperature

profiles, the normalized temperature self-correlation (〈T4〉/〈T〉4), the Planck-mean absorption co-

efficient and simplified Planck function (i.e.,T4) correlation (〈κPT4〉/〈κP〉〈T4〉), and the Planck-

mean absorption coefficient and incident radiation correlation (〈κPG〉/〈κP〉〈G〉). In the absence of

TRI, each of the latter two quantities would be equal to unity. The expression for temperature

self-correlation indicates that in the presence of thermalradiation there would always be TRI for

the level of turbulent fluctuations seen in reacting flows. The departures of each of the correla-

tions, stated above, from unity allow different contributions to TRI to be isolated and quantified.

In the remainder of the paper, the ˜ notation is dropped for clarity, and all quantities correspond to

resolved-scale values, unless noted otherwise.

A nondimensional optical thicknessκPL is introduced, whereL is an appropriate length scale.

Previous studies [24, 27, 28] have suggested that the fluctuations inκP (determined by local prop-

erties) and in incident radiationG (a nonlocal quantity) are uncorrelated (i.e.,〈κPG〉 ≈ 〈κP〉〈G〉) if

the mean free path for radiation is much larger than the turbulence eddy scale. This is the optically

thin fluctuation approximation, or OTFA (κPL 1). At the other extreme (κPL 1), the optical

thickness may be large compared to all hydrodynamic scales and chemical scales. In that case, the

fluctuations in intensity are generated locally and are expected to be correlated strongly with those

of the absorption coefficient. A diffusion approximation [26] can be used to model such cases. Be-

tween the two extremes are the cases where the smallest scales (Kolmogorov scales and/or flame

5

thickness) are optically thin while the largest (integral scales) are optically thick. Modeling of such

transitional cases is an outstanding challenge in TRI, and is a primary motivation for this study.

3 PROBLEM FORMULATION

Geometric Configuration

A statistically stationary and one-dimensional turbulentplanar channel flow is considered. Tur-

bulent channel flow is a classic configuration in the study of fluid dynamics, and has been the sub-

ject of numerous DNS and LES investigations [29–32]. This configuration has also been used to

study passive scalar statistics [33]. Moreover, a channel domain is a logical extension of classical

one-dimensional ‘slab’ problems often used in the study of radiation heat transfer.

The computational domain is a rectangular box with dimensions (2π, 2, π) in some unitH in

thex−, y− andz−directions, respectively (Fig. 1), with solid boundaries at the bottom (y=−H) and

top (y = +H). Mean-flow statistics vary only in the wall-normal direction (y). Periodic boundaries

are used in the streamwise (x) and the spanwise (z) directions for flow. The flow is driven by an

artificial streamwise body force that balances the frictionat the walls, thus allowing the pressure

to be periodic with zero mean pressure gradient in the computational domain. Two cases are

considered: a nonreacting case, and a reacting case.

Physical and Numerical Models

CFD Code OpenFOAM [34, 35], an open-source CFD code, has been used to perform the

large-eddy simulation for this configuration. The code has been validated by comparing computed

mean and rms velocity profiles with published DNS data from the literature for the same Reynolds

number (Reτ = 180, whereReτ is a Reynolds number based on the wall-friction velocity) [29].

This included a mesh-size and time step sensitivity study, and a comparison of results obtained in

the presence or absence of an explicit subfilter-scale (SFS)turbulence model [36]. A second study

confirmed that computed mean and rms passive scalar statistics are in good agreement with DNS

6

data [33]. While the code might not suffice for high-fidelity LES of near-wall turbulence dynamics,

it is quite satisfactory for generating a reasonable level of temperature and composition fluctuations

for the present purpose of studying thermal radiation and TRI. Quantitative comparisons between

LES and DNS mean and rms temperature (passive scalar) profiles are shown in Fig. 2. There is

excellent agreement between the LES mean temperature profiles and the DNS data. The mean

temperature profile has a steep gradient near the walls due tothe formation of turbulent boundary

layers and is relatively flat in the core of the channel as a result of turbulent mixing. The rms

profiles of temperature fluctuations are normalized by the wall friction temperature (Tτ), which is

defined as [33]

Tτ =qwρcPuτ

=kρcPuτ

d〈T〉dy

∣∣∣∣∣wall· (3)

Hereqw is the wall heat flux,cP is the specific heat,k is the thermal conductivity, anduτ is the

wall-friction velocity. The LES rms temperature profile in the absence of a subfilter-scale (SFS)

model is within 10% of the DNS data. The maximum temperature fluctuations are at the centerline

of the channel, in contrast to the near-wall maximum that is seen for the velocity fluctuations [29].

This is due to the nonzero mean temperature gradient at the centerline of the channel, which results

in the production of temperature fluctuations at that location.

Based on the above validation studies, all simulations reported here were performed with no ex-

plicit subfilter-scale model using second-order spatial (central differencing) and temporal (Crank-

Nicholson) discretizations. This corresponds to the Monotone Integrated LES (MILES) approach

for SFS modeling [37], where the SFS model is implicit in the numerical method. The finite-

volume grid used has 30, 60 and 50 cells in thex−, y− andz−directions, respectively, for the

nonreacting case and 60, 96 and 60 cells in thex−, y− andz−directions, respectively, for the re-

7

acting case. The computational time step corresponds to a maximum material Courant number of

0.3.

Physical Models An incompressible formulation is used. There is no feedbackof thermo-

chemistry to the hydrodynamics. The body force and fluid properties are set to achieve the desired

Reynolds number (Reτ = 186) [33], whereReτ is defined as

Reτ =uτHν, (4)

uτ is given by√τw/ρ, andτw is the shear stress at the wall. The kinematic viscosity,ν, is calculated

using Eq. (4) withuτ equal to unity. The wall-friction velocity,uτ, is also used to determine the

total wall friction and thus the magnitude of the streamwisebody force that is required to sustain

the flow.

For the nonreacting case, the top and bottom walls are isothermal at temperatures of 900 K and

1500 K, respectively (Fig. 1a). The laminar Prandtl number is set to 0.7. The channel boundaries

in the spanwise and streamwise directions are periodic. Thetop and bottom walls are treated as

black and diffuse for thermal radiation computations. The participatingmedium has a Planck-mean

absorption coefficient of the form

κp =Cκ[c0+c1

(AT

)+c2

(AT

)2+c3

(AT

)3+c4

(AT

)4+c5

(AT

)5](5)

where the coefficientsc0 to c5 andA have been taken from a radiation model suggested for water

vapor [38]. HereCκ is a coefficient that allows the optical thickness to be varied systematically

8

and independently of the other parameters. The values ofCκ used in this study are: 0.0005, 0.005,

0.05, 2.5, and 50. The Planck-mean absorption coefficient has an inverse temperature dependence

and it varies up to a factor of three over the temperature range of interest.

For the reacting case, the species fields are superposed on the flow field; i.e., the hydrodynamics

is solved as in the nonreacting case and the flow field is used totransport species through the

channel. The species and the energy variables are periodic only for the channel boundaries in the

spanwise direction; fixed-value inlet and zero-gradient outlet boundaries are used in the streamwise

directions for species and energy variables.

A nonpremixed system with one-step, irreversible, infinitely fast chemistry is considered:

F +O→ P (6)

A conserved scalar mixture fraction formulation is used where ξ = 0.0 for pure oxidizer,ξ = 1.0

for pure fuel, andξ = ξst= 0.5 for pure products. The Lewis number is set to unity.

A mixture-fraction profile is specified at the channel inlet,with ξ = 1 (pure fuel) near the upper

(y = +H) wall, ξ = 0 (pure oxidizer) near the lower (y = −H) wall, and a narrow transition zone

aroundy = 0:

ξ =12

(1+ tanh 15

y

H

)(7)

A zero-normal-gradient boundary condition is used for the mixture fraction at the channel out-

let, and at the top and the bottom walls. Species mass fractionsYF , YO, andYP are simple piecewise

linear functions ofξ, as shown in Fig. 3. In the absence of thermal radiation, the temperature also

9

is a piecewise linear function of mixture fraction, and has afunctional dependence exactly similar

to that for pure products. This piecewise linear function along with the inlet mixture-fraction pro-

file (Eq. 7) is used to calculate the temperature distribution at the inlet. In the presence of thermal

radiation, however, a separate energy (enthalpy) equationis solved. The specific absolute mixture

enthalpyh is:

h=∑

α

(hαYα+CpαYα [T −300]

)(8)

wherehα is the formation enthalpy of speciesα, Yα is its mass fraction,Cpα is its specific heat,

andT is the temperature. At the channel inlet, a profile for enthalpy is calculated from the above

expression and is assigned as the inlet boundary condition.The specific absolute enthalpy is as-

sumed to have a zero gradient at the outlet. The channel wallsare thermally insulated; there is zero

net heat flux to the wall:

qrad+qdiff = qrad−k∇T · n= 0 (9)

whereqrad is the net radiative heat flux to the wall,qdiff is the diffusive heat flux to the wall, and ˆn

is the outward unit-normal vector at the wall.

The RTE is solved for incident radiationG using a P1 spherical harmonics method, requiring

the solution of an elliptic PDE [26],

10

∇ · (Γ∇G) = −κP (4πIb−G) (10)

HereIb is the Planck function, andΓ is the optical diffusivity of the medium which for a nonscat-

tering medium, is given by

Γ =1

3κP(11)

For the reacting case the channel walls again are black and diffuse. The inlet is assumed to act

as a black wall at the local temperature. The channel outlet is assumed to be radiatively insulated,

with zero gradient for the incident radiation at the outlet boundary.

The participating medium has a Planck-mean absorption coefficient that is similar to Eq. (5),

but includes an explicit dependence on product mass fraction (YP):

κp =max

CκYP

[c0+c1

(AT

)+c2

(AT

)2+c3

(AT

)3+c4

(AT

)4+c5

(AT

)5], κPmin

(12)

whereκPmin is a very small absorption coefficient (∼ 0.01). The values ofCκ used in this study are:

0.0008, 0.008, 0.08, 0.16, 0.8, 4, and 16.

11

4 RESULTS AND DISCUSSION

For each configuration, results are presented for different optical thicknessesτ. Here τ is

defined as

τ =

〈κP〉y=0H, for the nonreacting case

∫ +H

−H〈κP〉dy

∣∣∣∣∣x=2πH

, for the reacting case

(13)

whereH is the half-channel height andy = 0 is the channel center-line location.

In each case, the initial flow field consisted of uniform streamwise velocity with high velocities

in the wall-normal and spanwise directions in a small regionat the core of the channel to promote

the production of turbulence. A statistically stationary turbulent flow field was established by al-

lowing the initial flow field to develop for approximately 100flow-through times. The temperature

(or mixture-fraction in the reacting configuration) field was introduced only after establishing the

statistically stationary turbulent flow field. The initial temperature field used in nonreacting con-

figuration was simply a linear profile between the two walls. The initial mixture-fraction field

used in reacting configuration was the inlet mixture-fraction profile of Eq. (7) propagated through

the entire channel. The temperature field (or mixture-fraction) was then allowed to evolve for ap-

proximately 40-50 flow-through times before switching on thermal radiation, and the system was

allowed to evolve for another 70-80 flow-through times to establish statistically stationary fields.

Averaging was performed over approximately 100 flow-through times to get proper time-averaged

statistics.

12

Configuration I: Nonreacting turbulent planar channel flow

For this statistically stationary and statistically one-dimensional configuration, mean quantities

correspond to averages over time and over planes parallel tothe walls. Computed mean tempera-

ture profiles in the absence of radiation and for different optical thicknesses are shown in Fig. 4.

As thermal radiation is introduced, initially (optically very thin medium,τ = 0.01) most of the

thermal radiative energy emitted by the hot wall penetratesthrough the medium and only a small

amount is absorbed. The mean temperature increases slightly from the no-radiation case, with the

departure increasing steadily from the hot wall to the cold wall. This can be explained from the

inverse-temperature-dependence of the Planck-mean absorption coefficient: the low-temperature

zones have a higher absorption coefficient. With an increase in optical thickness (τ = 0.1), most of

the thermal energy radiated by the hot wall still penetratesthrough the medium near the hot wall,

but is absorbed near the cold wall. In this case, the mean temperature profile deviates significantly

from the no-radiation case. As the optical thickness is increased further (τ = 1.0), the medium near

the hot wall also has a large absorption coefficient and absorbs some of the radiative energy emitted

by the hot wall. For this case, significant radiative energy still reaches the region near the cold wall

resulting in large temperature departures from the no-radiation case. A further increase in optical

thickness (τ = 50) results in most of the emitted thermal energy getting absorbed in the region near

the hot wall, and the departure from the no-radiation mean temperature distribution has a decreas-

ing trend as one moves from hot wall to cold wall. With furtherincrease in the optical thickness

(τ = 1250), the medium becomes essentially opaque and any radiation emitted is absorbed locally

close to the point of emission, and the mean temperature distribution approaches the no-radiation

temperature distribution. In the limit of large optical thickness, thermal radiation acts as a diffusive

process with a diffusivity proportional to 4σT3/3κR, whereκR is a Rosseland-mean absorption co-

efficient [26]. For largeκR the diffusivity due to thermal radiation is negligible, and the meanand

rms temperature profiles in the optically thick limit are thesame as those in the no-radiation case.

Both emission TRI and absorption TRI are negligible for thiscase (not shown). This is con-

13

sistent with findings from earlier studies for nonreacting turbulent flows [11, 39, 40] and can be

attributed to the relatively low level of temperature fluctuations (approximately 3% of the mean

temperatures) that are found in nonreacting flows.

Here the effect of varying optical thickness on a nonreacting high-temperature system has been

studied, for optical thicknesses ranging from very thin (τ = 0.01) to very thick (τ = 1250). For

comparison, we note that a 7 mm thick high-temperature region (T ∼ 1500K) of typical combus-

tion products (CO2 and H2O, at partial pressures of about 0.1 bar) would have an optical thickness

of approximately 0.01. On the other hand, a high-temperature region (T ∼ 1500K) about 50 m

thick with 10 ppm soot would correspond to an optical thickness of approximately 1000. Thus an

optical thickness of 1250 is extremely large, and serves here only to confirm that the model system

exhibits the correct behavior in the optically thick limit.In a practical system, an optical thickness

of approximately 15 or greater would correspond to an optically thick medium.

Configuration II: Chemically reacting nonpremixed turbule nt channel flow

For this configuration, mean quantities vary with bothx andy and, thus, mean quantities are

estimated by averaging over time and overz. A further averaging is performed about they = 0

plane to take advantage of the statistical symmetry of the problem and to reduce the statistical

errors in estimating mean quantities.

The mean temperature profiles at the exit of the computational domain (x = 2πH) are shown

in Fig. 5. The mean temperature peaks at the center of the channel, and drops to 300 K away

from the flame. When thermal radiation is considered the flameloses heat due to radiative energy

emission that heats up the adiabatic walls. Because of the heat loss from the flame, the flame core

temperature is decreased while the temperature of the gas adjacent to the heated walls increases

due to the formation of a thermal boundary layer. As the optical thickness increases, the flame

loses more heat resulting in higher wall temperatures and further reduction of the flame core mean

temperature. However, for very large optical thicknesses,emitted radiation is absorbed locally and

14

the mean temperature profile returns to that of the no-radiation case.

The rms temperature profiles atx= 2πH are shown in Fig. 6 for different optical thicknesses.

Turbulent flapping of the flame about the center-plane of the channel causes large variations in

temperature near the edges of the flame and gives rise to a double-peak profile. Thermal radiation

significantly alters the temperature fluctuations in the core of the flame.

The temperature self-correlation at the exit is shown in Fig. 7. The shape of the profile can be

attributed to the two-peak structure of the rms temperature. The location of the peak depends on

the level of temperature fluctuations relative to the mean temperature. Like the rms temperatures,

initially the temperature self-correlation decreases with increasing optical thickness and, once the

optical thickness attains a sufficiently high value, the temperature self-correlation increases again

with increasing optical thickness. The temperature self-correlation is found to be significant for all

optical thicknesses considered, due to the high level of temperature fluctuations in the presence of

chemical reactions.

Figure 8(a) presents the contribution to emission TRI of thecorrelation between Planck-mean

absorption coefficient and fourth power of temperature at thex = 2πH location. This contribu-

tion to emission TRI is found to be significant for all opticalthicknesses studied except for the

optically very thin case (τ = 0.02). The behavior and shape of the correlation can be explained

from the assumed functional dependence of the Planck-mean absorption coefficient on tempera-

ture andYP: κP is inversely proportional to temperature, but directly proportional toYP. Therefore,

the fluctuations in temperature andκP are negatively correlated at constantYP, and fluctuations in

product mass fraction andκP are positively correlated at constant temperature. These two opposing

phenomena compete with one another to determine the overallcorrelation. In the region near the

flame edge the temperature fluctuations have a dominant impact on κP fluctuations, thereby result-

ing in a correlation less than unity. Similarly, in regions slightly away from the flame edge,YP

fluctuations are dominant resulting in correlations much greater than unity. However, it is impor-

tant to realize that the region where theκP–T4 correlation is most significant has about two orders

15

of magnitude lower mean emission than the maximum mean emission, which occurs aty = 0, as

seen in Fig. 8(b). TheκP–T4 correlation is close to unity aty = 0, whereas at the same location

the temperature self-correlation is close to 2 as seen in Fig. 7. These numbers and the plots for

temperature self-correlation, theκP–T4 correlation and for mean heat emission indicate that here

the temperature self-correlation is the dominant contributor to emission TRI.

Absorption TRI atx= 2πH is shown in Fig. 9(a) for different optical thicknesses, and is seen

to be negligible for optically thin flames, consistent with the OTFA (optically thin fluctuation

approximation). Absorption TRI becomes significant at higher optical thicknesses; there thermal

radiation travels relatively short distances, and the local incident radiation has some correlation

with local emission and, therefore, with the local Planck-mean absorption coefficient. Regions

where theκP–G correlation is most significant have one-to-two orders of magnitude lower mean

absorption rates than the maximum mean absorption rate, as seen in Fig. 9(b).

The plots for temperature self-correlation, theκP–T4 correlation and theκP–G correlation

[Figs. 7, 8(a) and 9(a), respectively] reveal that, in the optically thin limit where τ = 0.02, the

temperature self-correlation is primarily the sole contributor to TRI, and it remains an important

contributor to TRI for the entire range of optical thicknessconsidered. This is because the temper-

ature self-correlation is significant in the region with large mean emission rates (i.e.,|y| ≤ 0.3) as

opposed to theκP–T4 correlation and theκP–G correlation which are dominant in the regions with

mean rates one or two orders of magnitude lower than the maximum mean emission and mean

absorption rates, respectively (see Figs. 8(b) and 8(b)). TheκP–T4 correlation is important at all

optical thicknesses except in the optically thin limit, whereas absorption TRI (κP–G correlation)

is important only in the optically intermediate to thick regions. Note that for very large optical

thicknesses both emission TRI and absorption TRI are strong, but their net effect is small due

to cancellation (Figs. 8 and 9): in the optically thick limitlocal values ofG approach 4πIb and,

therefore, the fluctuations inG are directly proportional to the fluctuations inT4. Absorption TRI

[Fig. 9(a)] has similar shape and behavior as emission TRI [Fig. 8(a)] for very large optical thick-

16

nesses. Similarly, it can be seen from the mean emission and mean absorption plots [Figs. 8(b)

and 9(b)] that the mean emission rate is identical to the meanabsorption rate for very large optical

thicknesses, resulting in mean temperatures approaching those of the no-radiation case as seen in

Fig. 5.

5 CONCLUSIONS

Large-eddy simulation of incompressible turbulent planarchannel flow has been conducted,

including thermal radiation. Cases without and with chemical reaction were considered with sys-

tematic variation in optical thickness. In the absence of chemical reactions radiation significantly

modifies the mean temperature fields, but TRI were found to be negligible. The no-radiation results

are recovered in both the optically thin and optically thicklimits. With chemical reactions emission

TRI is important at all optical thicknesses, while absorption TRI increases with increasing optical

thickness. For the present configuration the temperature self-correlation makes the most important

contribution to emission TRI. The other noted contributionto emission TRI is Planck-mean ab-

sorption coefficient–Planck function correlation. Absorption TRI is as significant as emission TRI

at large optical thicknesses, and the two tend to cancel eachother out.

Because of the number of simplifications made in this analysis, results from this idealized case

should be considered as qualitative. The proposed next steps are: 1) finite-rate chemistry with a

filtered density function (FDF) method to accommodate turbulent–chemistry interactions (TCI);

2) nongray thermal radiation properties; and 3) a more accurate radiative transfer equation (RTE)

solver such as photon Monte Carlo (PMC) method or a higher-orderP−N method.

ACKNOWLEDGMENT

This work has been supported by the National Science Foundation [Grant Number CTS-0121573]

and NASA [Award Number NNX07AB40A].

17

References

[1] H. Pitsch, Improved pollutant predictions in large-eddy simulations of turbulent non-

premixed combustion by considering scalar dissipation rate fluctuations,Proceedings of The

Combustion Institute29 (2002), 1971–1978.

[2] M. R. H. Sheikhi, T. G. Drozda, P. Givi, F. A. Jaberi and S. B. Pope, Large eddy simulation

of a turbulent nonpremixed piloted methane jet flame (SandiaFlame D),Proceedings of The

Combustion Institute30 (2005), 549–556.

[3] P. Moin and S. V. Apte, Large-eddy simulation of realistic gas turbine combustors,AIAA

Paper No. AIAA-2004-330(2004).

[4] S. James, J. Zhu and M. S. Anand, Large-eddy simulations of gas turbine combustors, AIAA

Paper no. 2005-0552 (2005).

[5] P. Flohr, CFD modeling for gas turbine combustors, InTurbulent Combustion(Edited by L.

Vervisch, D. Veynante and J. P. A. J. Van Beeck), von Karman Institute for Fluid Dynamics

Lecture Series 2005-02, Rhode-Saint-Genese, Belgium March 2005.

[6] I. Celik, I. Yavuz and A. Smirnov, Large-eddy simulations of in-cylinder turbulence for IC

engines: A review,Internl. J. Engine Res.2 (2001), 119–148.

[7] D. C. Haworth, A review of turbulent combustion modelingfor multidimensional in-cylinder

CFD,SAE Transactions, Journal of Engines(2005), 899–928.

[8] S. Richard, O. Colin, O. Vermorel, A. Benkenida, A. Angelberger and D. Veynante, Towards

large-eddy simulation of combustion in spark-ignition engines,Proceedings of the Combus-

tion Institute31 (2007), 3059–3066.

[9] A. A. Townsend, The Effects of Radiative Transfer on Turbulent Flow of a Stratified Fluid, J.

Fluid Mech.4 (1958), 361–375.

[10] T.H. Song and R. Viskanta, Interaction of Radiation with Turbulence: Application to a Com-

bustion System,J. Thermoph. Heat Transfer1 (1987), 56–62.

[11] A. Soufiani, P. Mignon and J. Taine, Radiation–Turbulence Interaction in Channel Flows of

18

Infrared Active Gases, InProceedings of the International Heat Transfer Conference, Vol. 6,

pp. 403–408, ASME 1990.

[12] J. P. Gore and G. M. Faeth, Structure and spectral radiation properties of turbulent ethylene/air

diffusion flames, InTwenty-First Symposium (International) on Combustion, pp. 1521–1531,

The Combustion Institute 1986.

[13] J. P. Gore and G. M. Faeth, Structure and Spectral Radiation Properties of Luminous Acety-

lene/Air Diffusion Flames,J. Heat Transfer110 (1988), 173–181.

[14] M. E. Kounalakis, J. P. Gore and G. M. Faeth, Turbulence/Radiation Interactions in Non-

premixed Hydrogen/Air Flames, InTwenty-Second Symposium (International) on Combus-

tion, pp. 1281–1290, The Combustion Institute 1988.

[15] M. E. Kounalakis, Y. R. Sivathanu and G. M. Faeth, Infrared Radiation Statistics of Nonlu-

minous Turbulent Diffusion Flames,J. Heat Transfer113(2) (1991), 437–445.

[16] G. M. Faeth, J. P. Gore, S. G. Shuech and S. M. Jeng, Radiation from Turbulent Diffusion

Flames,Ann. Rev. Numer. Fluid Mech. Heat Trans.2 (1989), 1–38.

[17] P. J. Coelho, Detailed Numerical Simulation of Radiative Transfer in a Nonluminous Turbu-

lent Jet Diffusion Flame,Comb. Flame136 (2004), 481–492.

[18] L. Tesse, F. Dupoirieux and J. Taine, Monte Carlo Modeling of Radiative Transfer in a Tur-

bulent Sooty Flame,Int. J. Heat Mass Transfer47 (2004), 555–572.

[19] S. Mazumder and M. F. Modest, A PDF Approach to Modeling Turbulence–Radiation Inter-

actions in Nonluminous Flames,Int. J. Heat Mass Transfer42 (1999), 971–991.

[20] G. Li and M. F. Modest, Application of Composition PDF Methods in the Investigation of

Turbulence–Radiation Interactions,J. Quant. Spectrosc. Radiat. Transfer73(2–5) (2002),

461–472.

[21] Y. Wu, D. C. Haworth, M. F. Modest and B. Cuenot, Direct Numerical Simulation of Turbu-

lence/Radiation Interaction in Premixed Combustion Systems,Proceedings of the Combus-

tion Institute30 (2005), 639–646.

19

[22] K. V. Deshmukh, D. C. Haworth and M. F. Modest, Direct Numerical Simulation of Turbu-

lence/Radiation Interactions in Nonpremixed Combustion Systems, Proceedings of the Com-

bustion Institute31 (2007), 1641–1648.

[23] A. J. Chandy, D. J. Glaze and S. H. Frankel, Mixing Modelsand Turbulence-Radiation In-

teractions in Non-Premixed Jet Flames via the LES/FMDF Approach, In5th US Combustion

Meeting, San Diego, CA, March 25-28, 2007.

[24] V. P. Kabashnikov and G. I. Kmit, Influence of Turbulent Fluctuations on Thermal Radiation,

J. Applied Spectroscopy31 (2) (1979), 963–967.

[25] P. Coelho, Numerical Simulation of the Interaction Between Turbulence and Radiation in

Reactive Flows,Progress in Energy and Combustion Science33 (2007), 311–383.

[26] M. F. Modest,Radiative Heat Transfer(2nd Edn), Academic Press, New York 2003.

[27] V. P. Kabashinikov, Thermal Radiation of Turbulent Flows in the Case of Large Fluctuations

of the Absorption Coefficient and the Planck Function,J. Eng. Phys.49(1) (1985), 778–784.

[28] V. P. Kabashinikov and G. I. Myasnikova, Thermal Radiation in Turbulent Flows—

Temperature and Concentration Fluctuations,Heat Transfer-Soviet Research17(6) (1985),

116–125.

[29] R. D. Moser, J. Kim and N. N. Mansour, Direct Numerical Simulation of Turbulent Channel

Flow up toReτ=590,Physics of Fluids11 (4) (1999), 943–945.

[30] K. Horiuti, The Role of the Bardina Model in Large Eddy Simulation of Turbulent Channel

Flow, Physics of Fluids A1 (2) (1989), 426–428.

[31] U. Piomelli, P. Moin and J. H. Ferziger, Model Consistency in Large Eddy Simulation of

Turbulent Channel Flows,Physics of Fluids31 (1988), 1884–1891.

[32] P. Moin and J. Kim, Numerical Investigation of Turbulent Channel Flow,J. Fluid Mech.118

(1982), 341–377.

[33] B. Debusschere and C. J. Rutland, Turbulent Scalar Transport Mechanisms in Plane Channel

and Couette Flows,Int. J. Heat Mass Transfer47 (2004), 1771–1781.

20

[34] H. G. Weller, G. Tabor, H. Jasak and C. Fureby, A Tensorial Approach to Continuum Me-

chanics Using Object Oreinted Techniques,Computers in Physics12 (6) (1998), 620–631.

[35] Seewww.openfoam.org, January, 2006.

[36] A. Gupta, Large-Eddy Simulation of Turbulence/Chemistry/Radiation Interactions in Planar

Channel Flow, Master’s thesis, The Pennsylvania State University, University Park, PA 2007.

[37] J. P. Boris, F. F. Grinstein, E. S. Oran and R. J. Kolbe, New Insights Into Large Eddy Simula-

tion, Fluid Dynamics Research10 (1992), 199–228.

[38] Sandia National Laboratories Combustion Research Facility, Intern’l. Workshop on Measure-

ment and Computation of Turbulent Nonpremixed Flames2002.

[39] S. Mazumder and M. F. Modest, Turbulence–Radiation Interactions in Nonreactive Flow of

Combustion Gases,J. Heat Transfer121 (1999), 726–729.

[40] V. Singh, Study of Turbulence Radiation Interactions Using LES of Planar Channel Flow,

Master’s thesis, The Pennsylvania State University, University Park, PA 2005.

21

List of Figures

1 Instantaneous temperature contours for (a) nonreacting flow and (b) reacting flow. . 23

2 Computed mean and rms passive scalar profiles, and their comparison with DNS

of Debusschere and Rutland [33]. . . . . . . . . . . . . . . . . . . . . . . .. . . . 24

3 Species mass fractions as a function of mixture fraction,ξ. . . . . . . . . . . . . . 25

4 Computed mean temperature profiles with variations in optical thickness for the

nonreacting case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26

5 Computed mean temperature profiles with variations in optical thickness for the

reacting case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Computed rms temperature profiles with variations in optical thickness for the re-

acting case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Temperature self-correlation for several optical thicknesses for the reacting case. . 29

8 Correlation betweenκP andT4 for several optical thicknesses for the reacting case. 30

9 Absorption TRI for several optical thicknesses for the reacting case. . . . . . . . . 31

22

(a) Nonreacting Flow (b) Reacting Flow

Figure 1. Instantaneous temperature contours for (a) nonreacting flow and (b) reacting flow.

23

y / H<

T>

<T

’2 >1

/2/T

τ

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

2

2.5

3

3.5

DNSSFS Model only for flowNo SFS Model

Figure 2. Computed mean and rms passive

scalar profiles, and their comparison with DNS

of Debusschere and Rutland [33].

24

Mixture Fraction ( ξ)

Mas

sF

ract

ion

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

YOYP

YF

Figure 3. Species mass fractions as a func-

tion of mixture fraction, ξ.

25

y / H

Mea

nT

empe

ratu

re(K

)

-1 -0.5 0 0.5 1900

1000

1100

1200

1300

1400

1500

No RadiationOptically very thin, τ = 0.01Optically thin, τ = 0.1Optically intermediate, τ = 1.0Optically thick, τ = 50Optically very thick, τ = 1250

Figure 4. Computed mean temperature profiles with varia-

tions in optical thickness for the nonreacting case.

26

O O O O O O O O O O O O

O

O

O

O

O

y / H

Mea

nT

empe

ratu

re(K

)

-1 -0.8 -0.6 -0.4 -0.2 0200

400

600

800

1000

1200

No Radiationτ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

Figure 5. Computed mean temperature profiles with varia-

tions in optical thickness for the reacting case.

27

O O O O O O O O OO

O

O

O

O

O

O

O

y / H

RM

ST

empe

ratu

re(K

)

-1 -0.8 -0.6 -0.4 -0.2 00

200

400

No Radiationτ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

Figure 6. Computed rms temperature profiles with variations

in optical thickness for the reacting case.

28

O O O O O O O O O OO

O

O

O

O

O

O

y / H

⟨T4

⟩/⟨T

⟩4

-1 -0.8 -0.6 -0.4 -0.2 00

2

4

6

8

No Radiationτ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

Figure 7. Temperature self-correlation for several optical

thicknesses for the reacting case.

29

O O O O OO

O

O

O

O

OO

O

O

OO O

y / H

⟨κPT

4 ⟩/⟨κ

P⟩⟨

T4 ⟩

-1 -0.8 -0.6 -0.4 -0.2 00

0.5

1

1.5

2

2.5

3

3.5τ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

(a) Normalized correlation

O O O OO

O

O

O

O

O

O

O

O

O

O

OO

y / H

4σ⟨κ

PT

4 ⟩(in

J/m

3 )

-1 -0.8 -0.6 -0.4 -0.2 0101

102

103

104

105

106

107

108

τ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

(b) Mean radiative heat emission rate per unitvolume

Figure 8. Correlation between κP and T4 for several optical thicknesses for the reacting case.

30

O O O O OO

O

O

O

O

OO

O

O

OO

O

y / H

⟨κPG

⟩/⟨κ

P⟩⟨

G⟩

-1 -0.8 -0.6 -0.4 -0.2 00

0.5

1

1.5

2

2.5

3τ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

(a) Absorption TRI

O O O OO

O

O

O

O

O

O

O

O

O

O

OO

y / H

⟨κPG

⟩(in

J/m

3 )

-1 -0.8 -0.6 -0.4 -0.2 0101

102

103

104

105

106

107

108

τ = 0.02τ = 0.1τ = 0.85τ = 1.6τ = 6.5τ = 37τ = 165O

(b) Mean radiative heat absorption rate per unitvolume

Figure 9. Absorption TRI for several optical thicknesses for the reacting case.

31